Let $\pi:S\to \Bbb{P}^1$ be a minimal elliptic fibration. I'm trying to understand why the following is true:
S is rational if and only $\chi_\text{top}(S)=12$.
In Schütt and Shioda's Elliptic Surfaces page 46 there is an indication for $(\Rightarrow)$: by minimality, $K_S^2=0$ and by Noether's formula $\chi_\text{top}(S)=12(K_S^2+\chi(S))=12\chi(S)$, so it remais to show that $\chi(S)=1$. He uses the fact that $K_S=(\chi(S)-2)F$ for a general fibre $F$ and that $K_S$ is anti-ample and concludes directly that $\chi(S)=1$. I'm not familiar with ample divisors, but as far as I could read about it, I don't understand why that follows.
For the converse $(\Leftarrow)$, I suppose the idea is to argue that $K_S=-F$ implies $K_S$ is anti-ample, but again it's not clear to me why.
Thank you for your help!